Question

question 1 points 3 the second and third terms of a geometric series are 128 and 96 respectively. find the first term. 170.67 96 128 256

Explanation:

Step1: Recall Geometric Sequence Formula

In a geometric sequence, the \(n\)-th term is \(a_n = a_1r^{n - 1}\), where \(a_1\) is the first term and \(r\) is the common ratio. The second term \(a_2=a_1r\) and the third term \(a_3=a_1r^{2}\). Given \(a_2 = 128\) and \(a_3=96\), we can find \(r\) by \(\frac{a_3}{a_2}\).
\(r=\frac{96}{128}=\frac{3}{4}\)

Step2: Find the First Term \(a_1\)

Since \(a_2=a_1r\), we can solve for \(a_1\) by \(a_1=\frac{a_2}{r}\). Substitute \(a_2 = 128\) and \(r=\frac{3}{4}\) (wait, no, wait: Wait, if \(a_2 = 128\) and \(a_3 = 96\), then \(r=\frac{a_3}{a_2}=\frac{96}{128}=\frac{3}{4}\)? Wait, no, wait, maybe I mixed up the terms. Wait, the problem says "the second and third terms of a geometric series are 128 and 96 respectively". So \(a_2 = 128\), \(a_3=96\). Then \(r=\frac{a_3}{a_2}=\frac{96}{128}=\frac{3}{4}\). Then \(a_1=\frac{a_2}{r}=\frac{128}{\frac{3}{4}}=\frac{128\times4}{3}=\frac{512}{3}\approx170.67\). Wait, but let's check the options. One of the options is 170.67. So that's the first term.

Answer:

170.67